This is joint work with Angus Macintyre. We study model-theoretic properties ofthe ring of adeles, the hyperring of adele classes (studied by Connes-Consani),and residual hyperfields of valued fields (in the sense of Krasner).

I shall present a geometric property valid in many Hrushovskiamalgamation constructions, relative CM-triviality, and deriveconsequences on definable groups: modulo their centre they are alreadyproducts of groups interpretable in the initial theories used for theconstruction. For the bad field constructed in this way, I shallmoreover classify all interpretable groups up to isogeny.